Talk:Normal polytope

The article talks about "unimodular triangulation" without defining it. What does it mean? --82.181.119.60 (talk) 18:35, 14 December 2014 (UTC)

Normal vs normal

This page is inconsistent. In the first paragraph it says

" a convex lattice polytope P is called normal if it has the following property: given any positive integer n, every lattice point of the dilation nP, obtained from P by scaling its vertices by the factor n and taking the convex hull of the resulting points, can be written as the sum of exactly n lattice points in P "

While in the "Definition" section, normality becomes

"$c\in \mathbb {N} ,z\in cP\cap L\implies \exists x_{1},\ldots ,x_{c}\in P\cap L$ such that $x_{1}+\cdots +x_{c}=z$."

(L is the lattice spanned by the lattice points in P). This is not the same, since $cP\cap L$ can have less lattice points than $cP\cap \mathbb {Z} ^{d}$. Try it out on the convex hull of the points $[0,0,0],[1,1,0],[1,0,1],[0,1,1]$. This is not normal in the sense of the introduction, but in the sense of the definition stated here.

It seems that what is called "normal" in the introduction, becomes "integrally closed" in the definition. Digging deeper, there really seem to be two mathematical communities here fighting over what normality should be. The Miller Sturmfels reference for example agrees with the introduction. However, Bruns and Gubeladze agree with the definition. In fact, the definition was added in 2009 and this is almost the same time as articles from Gubeladze pop up with the exact same definition. See for example http://math.sfsu.edu/gubeladze/publications/fpsac2010.pdf. Another article by Sam Payne states right at the beginning that normality should be the introduction variant, while then going on referencing work by Gubeladze, who is using the definition variant.

To summarize: Wtf? I do not know or decide which variant is correct, but at least this wikipedia page should be consistent. Maybe state clearly that there are two different variants of "normal polytope"? What is the procedure here? — Preceding unsigned comment added by 130.149.13.123 (talk) 09:26, 23 November 2018 (UTC)